The projective tensor product $\ell_1\widehat{\otimes}X$ is naturally isometrically isomorphic to the $\ell_1$-sum of countably many copies of $X$. The uncompleted tensor product $\ell_1 \odot X$ is then the linear span of elements of the form $(\xi_n x)$, where $(\xi_n)$ is in $\ell_1$ under this identification, which is hardly complete as there exist infinite convergent series.

In general the proof goes along the same lines -- it uses the possibility of (non-unique) representation of elements of the projective tensor product as infinite series of simple tensors. You then have to show that if $X$ and $Y$ are infinite-dimensional then there is an infinite series that cannot be truncated to a finite one.

The projective tensor product $\ell_1\widehat{\otimes}X$ is naturally isometrically isomorphic to the $\ell_1$-sum of countably many copies of $X$. The uncompleted tensor product $\ell_1 \odot X$ is then the linear span of elements of the form $(\xi_n x)$, where $(\xi_n)$ is in $\ell_1$ under this identification, which is hardly complete as there exist infinite convergent series. For example, take a linearly independent sequence $(x_n)_{n=1}^\infty$ of unit vectors in $X$ and consider $(n^{-2}x_n)_{n=1}^\infty$; it does not belong to (the image of) $\ell_1\odot X$.

In general the proof goes along the same lines -- it uses the possibility of (non-unique) representation of elements of the projective tensor product as infinite series of simple tensors. You then have to show that if $X$ and $Y$ are infinite-dimensional then there is an infinite series that cannot be truncated to a finite one.

The projective tensor product $\ell_1\widehat{\otimes}X$ is naturally isometrically isomorphic to the $\ell_1$-sum of countably many copies of $X$. The uncompleted tensor product $\ell_1 \odot X$ is then the algebraic direct sum of countably copies of $X$ under this identification, which is hardly complete as there exist infinite convergent series.

In general the proof goes along the same lines -- it uses the possibility of (non-unique) representation of elements of the projective tensor product as infinite series of simple tensors. You then have to show that if $X$ and $Y$ are infinite-dimensional then there is an infinite series that cannot be truncated to a finite one.

The projective tensor product $\ell_1\widehat{\otimes}X$ is naturally isometrically isomorphic to the $\ell_1$-sum of countably many copies of $X$. The uncompleted tensor product $\ell_1 \odot X$ is then the linear span of elements of the form $(\xi_n x)$, where $(\xi_n)$ is in $\ell_1$ under this identification, which is hardly complete as there exist infinite convergent series.

In general the proof goes along the same lines -- it uses the possibility of (non-unique) representation of elements of the projective tensor product as infinite series of simple tensors. You then have to show that if $X$ and $Y$ are infinite-dimensional then there is an infinite series that cannot be truncated to a finite one.